Exercise 7, § 4

a) Let l be an integer $\geq 1$ and n \in \mathbf{N}. Show that there exists a unique decreasing sequence of integers: $a_{l,l}(n) \geq a_{l,l-1}(n) \geq \cdots \geq a_{l,1}(n) \geq 0$ such that

Show that one has $n \leq m$ if and only if $(a_{l,l}(n), a_{l,l-1}(n), ..., a_{l,1}(n))$ is less than $(a_{l,l}(m), ..., a_{l,1}(m))$ for the lexicographic order. Set

and for $l \geq 2$, denote by $\partial_l^*(n)$ the smallest integer p such that $\partial_{l-1}(p) \geq n$. Calculate $a_{l+1,i}(\partial_l(n)), a_{l-1,i}(\partial_l^*(n))$ in terms of the $a_{l,i}(n)$. b) We say that a mapping H : \mathbf{N} \to \mathbf{N} is a Macaulay function if

Show that it comes to the same thing as saying that

Let $H : \mathbf{N} \to \mathbf{N}$ be a Macaulay function. Show that only two cases are possible: α) There exists $r \in \mathbf{N}$ such that $H(j) = 0$ for all $j \geq r$. β) There exist $r \in \mathbf{N}$ and a finite sequence of integers

such that for all $j \geq r$, one has Show that $r$ and the sequence $b_0 \geq b_1 \geq \cdots \geq b_r \geq 0$ are uniquely determined by the function $H$. We set $d(H) = 0$ in case α) and $d(H) = b_0 + 1$ in case β), so that $d(H) - 1$ is the degree of the polynomial () in the variable $j$. Let $a(H) \frac{j^{d(H)-1}}{(d(H)-1)!}$ be the term of highest degree of the polynomial (). Show that $a(H)$ is an integer $> 0$. Show that a polynomial in one variable $j$ with coefficients in $\mathbf{Q}$, which takes integer values on the integers and which is strictly positive for $j$ sufficiently large, is not necessarily of the form (*). c) Let $H : \mathbf{N} \to \mathbf{N}$ be a Macaulay function. We set so that $S_H(T) \in \mathbf{Z}[[T]]$. Show that $S_H(T) \in \mathbf{Z}[T]$ or else that there exist a finite sequence of integers uniquely determined by $H$, and a polynomial $P \in \mathbf{Z}[T]$ such that one has Show that $c_0 = d(H)$ and that $a(H)$ is the number of integers $c_i$ such that $c_i = d(H)$. The rational fraction $\sum_{n=0}^r \frac{T^n}{(1-T)^{c_n}}$ is called the asymptotic expansion of $H$. Let $A_1, ..., A_d$ be integers, with $A_d \neq 0$, such that the rational fraction is a polynomial. Show that the sequence $(A_n)_{1 \leq n \leq d}$ uniquely determines and is uniquely determined by the asymptotic expansion of $H$. In particular one has $d = d(H), A_d = a(H)$. Examine the case $d(H) = 2$ in greater detail. d) Let $H : \mathbf{N} \to \mathbf{N}$ be a Macaulay function. Show that the following conditions are equivalent: (i) $\partial_l^* H(l) = H(l-1)$ for all $l \geq 2$, (ii) $H$ is the smallest of the Macaulay functions having the same asymptotic expansion as $H$. Such Macaulay functions are said to be extreme. Let $c_0 \geq c_1 \geq \cdots \geq c_r > 0$ be a sequence of integers. Show that there exists one and only one extreme Macaulay function having $\sum_{n=0}^r \frac{T^n}{(1-T)^{c_n}}$ as asymptotic expansion. Show that, if $H$ is an extreme Macaulay function, one has $H(1) = d(H)$ or else $H(1) = d(H) + 1$. Deduce that for every Macaulay function $H$, one has $H(1) \geq d(H)$. e) Let $H : \mathbf{N} \to \mathbf{N}$ be a Macaulay function. Show that the following properties are equivalent: (i) one has $S_H(T) = \frac{1}{(1-T)^{d(H)}}$,

(ii) $H(1) = d(H)$, (iii) $\partial_l H(l) = H(l + 1)$ for all $l \geqslant 1$. In particular, every function such that $H(1) = d(H)$ is extreme.